The van der Waals Interaction Derek Stampone Binghamton University - - PowerPoint PPT Presentation

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The van der Waals Interaction Derek Stampone Binghamton University - - PowerPoint PPT Presentation

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The van der Waals Interaction Derek Stampone The van der Waals Interaction Derek Stampone Binghamton University 5/9/2013 Outline The van der Waals Interaction 1 Introduction Derek Stampone 2 The Hamiltonian Outline Introduction The

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The van der Waals Interaction Derek Stampone

The van der Waals Interaction

Derek Stampone

Binghamton University

5/9/2013

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The van der Waals Interaction Derek Stampone Outline Introduction The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion

Outline

1 Introduction 2 The Hamiltonian 3 Semi-Classical Approach 4 Second-Order Perturbation 5 Van der Waals Interactions: Evaluations by use of a

statistical mechanical method

6 Conclusion

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The van der Waals Interaction Derek Stampone Outline Introduction The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion

The van der Waals Interaction

Interaction between neutral objects

1 Keeson Force - Force between

two permanent dipoles

2 Debeye Force - Force between

permanent dipole and induced dipole

3 London Dispersion Force - Force

between two induced dipoles.

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The van der Waals Interaction Derek Stampone Outline Introduction The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion

System Arrangement

"e! +e! x1! "e! +e! x2! R! Figure: Two nearby polarizable atoms.

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The van der Waals Interaction Derek Stampone Outline Introduction The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion

Unperturbed Hamiltonian

H0 = p2

1

2m + 1 2kx2

1 + p2 2

2m + 1 2kx2

2

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The van der Waals Interaction Derek Stampone Outline Introduction The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion

Coulomb Interaction

H′ = 1 4πǫ0 e2 R − e2 R − x1 − e2 R + x2 + e2 R − x1 − x2

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The van der Waals Interaction Derek Stampone Outline Introduction The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion

Perturbed Hamiltonian

Take the limit as x1 ≪ R and x ≪ R we get H′ ≈ − e2x1x2 2πǫ0R3

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The van der Waals Interaction Derek Stampone Outline Introduction The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion

Change of Variables

x± = 1 √ 2(x1 ± x2) p± 1 √ 2(p1 ± p2)

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The van der Waals Interaction Derek Stampone Outline Introduction The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion

Change of Variables

x± = 1 √ 2(x1 ± x2) p± 1 √ 2(p1 ± p2)

H = 1 2mp2

+ + 1

2

  • k −

e2 2πǫ0R3

  • x2

+

  • +

1 2mp2

− + 1

2

  • k +

e2 2πǫ0R3

  • x2

−

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The van der Waals Interaction Derek Stampone Outline Introduction The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion

Change of Variables

x± = 1 √ 2(x1 ± x2) p± 1 √ 2(p1 ± p2)

H = 1 2mp2

+ + 1

2

  • k −

e2 2πǫ0R3

  • x2

+

  • +

1 2mp2

− + 1

2

  • k +

e2 2πǫ0R3

  • x2

−

  • E = 1

2(ω+ + ω−) ω± =

  • k ∓ (e2/2πǫ0R3)

m

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The van der Waals Interaction Derek Stampone Outline Introduction The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion

Without H′, E0 = ω ω =

  • k/m
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The van der Waals Interaction Derek Stampone Outline Introduction The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion

Without H′, E0 = ω ω =

  • k/m

We’ll assume that k ≫ (e2/2πǫ0R3), and get ∆V ≡ E − E0 ≈ −

  • 8m2ω3

e2 2πǫ0 2 1 R6

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The van der Waals Interaction Derek Stampone Outline Introduction The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion

Second-Order Perturbation

E2

0 = ∞

  • n=1

|ψn|H′|ψ0|2 E0 − En

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The van der Waals Interaction Derek Stampone Outline Introduction The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion

Second-Order Perturbation

E2

0 = ∞

  • n=1

|ψn|H′|ψ0|2 E0 − En where |ψ0 = |0|0 |ψn = |n1|n2 and H′ ≈ − e2x1x2 2πǫ0R3

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The van der Waals Interaction Derek Stampone Outline Introduction The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion

Second-Order Perturbation (Cont.)

Write x in terms of raising and lowering operators. Only non-zero term will be n1 = n2 = 1

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The van der Waals Interaction Derek Stampone Outline Introduction The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion

Second-Order Perturbation (Cont.)

Write x in terms of raising and lowering operators. Only non-zero term will be n1 = n2 = 1 E2

0 =

  • e2

2πǫ0R3 2 |1|x|0|2|1|x|0|2 1

2ω0 + 1 2ω0

  • −

3

2ω0 + 3 2ω0

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The van der Waals Interaction Derek Stampone Outline Introduction The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion

Second-Order Perturbation (Cont.)

Write x in terms of raising and lowering operators. Only non-zero term will be n1 = n2 = 1 E2

0 =

  • e2

2πǫ0R3 2 |1|x|0|2|1|x|0|2 1

2ω0 + 1 2ω0

  • −

3

2ω0 + 3 2ω0

  • E2

0 = −

  • 8m2ω3

e2 2πǫ0 2 1 R6

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The van der Waals Interaction Derek Stampone Outline Introduction The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion

Van der Waals Interactions: Evaluations by use of a statistical mechanical method

Johan S. Høye’s paper is about using statistical mechanical methods to show an equivalence between the Casimir effect and second-order perturbation to Schr¨

  • dinger’s equation.
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The van der Waals Interaction Derek Stampone Outline Introduction The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion

Van der Waals Interactions: Evaluations by use of a statistical mechanical method

Johan S. Høye’s paper is about using statistical mechanical methods to show an equivalence between the Casimir effect and second-order perturbation to Schr¨

  • dinger’s equation.

Høye reduces the problem between interacting dipoles to a classical polymer problem in four dimensions with imaginary

  • time. He also avoids quantization of the electromagnetic field.
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The van der Waals Interaction Derek Stampone Outline Introduction The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion

Conclusion

The van der Waals interaction is a weak but ubiquitous force that can exist in a variety of situations. The weak attraction between dipoles is an important result. The van der Waals interaction is a framework to describe a variety of phenomenon including the Casimir effect.

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The van der Waals Interaction Derek Stampone Outline Introduction The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion

Conclusion

Research has been done on how geckos use the van der Waals interaction to cling to surfaces and how to apply similar methods to produce sticky tape.